Analysis method of modulated thermogravimetry, also known as modulated-temperature thermogravimetry, was introduced by R. L. Blaine and B. K. Hahn, Obtaining Kinetic Parameters by modulated Thermogravimetry, Journal of Thermal Analysis, Vol. 54 (1998) 695-704. It is used now widely as the standard method of modulated thermogravimetric analysis. It is described e.g. in Roger L. Blaine, U.S. Pat. No. 6,336,741 (2002).
Firstly, we will shortly put here the main idea of this standard method. The thermogravimetric measurements must be done under modulated temperature conditions, where the temperature is the sum of an underlying linear heating and temperature oscillations. The amplitude of the temperature oscillations is usually from 5K to 10K, which is much higher than for modulated DSC (cf. e.g. M. Reading, B. K. Hahn and B. S. Crow, U.S. Pat. No. 5,224,775 (1993)), where the typical temperature amplitude is about 0.5K. Period is usually from 60 to 300 s, underlying heating rate is usually from 1 to 20K/min.
The main kinetic equation is
                                          ⅆ            α                                ⅆ            t                          =                              Zf            ⁡                          (              α              )                                ⁢                      exp            ⁡                          (                              -                                  Ea                  RT                                            )                                                          (        1        )            wherein α is the degree of conversion, t is the time, Z is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the (absolute) temperature.
A list of these and other notations used in this description can be found at the end of the description.
If equation (1) is written twice for the same chemical process but for different conversion values α1 and α2 and correspondent temperatures T1 and T2 then the following equations for each reaction rate can be written:
                                          ⅆ                          α              1                                            ⅆ            t                          =                              Zf            ⁡                          (                              α                1                            )                                ⁢                      exp            ⁡                          (                              -                                  Ea                                      RT                    1                                                              )                                                          (        2        )                                                      ⅆ                          α              2                                            ⅆ            t                          =                              Zf            ⁡                          (                              α                2                            )                                ⁢                      exp            ⁡                          (                              -                                  Ea                                      RT                    2                                                              )                                                          (        3        )            
Then the equations are divided by each other and the logarithm is taken:
                              ln          ⁡                      (                                          ⅆ                                  α                  1                                                            ⅆ                                  α                  2                                                      )                          =                              ln            ⁡                          (                                                f                  ⁡                                      (                                          α                      1                                        )                                                                    f                  ⁡                                      (                                          α                      2                                        )                                                              )                                -                                    Ea              R                        ⁢                          (                                                1                                      T                    1                                                  -                                  1                                      T                    2                                                              )                                                          (        4        )            
From this expression the activation energy can be found:
                    Ea        =                  R          *                      (                                          ln                ⁡                                  (                                                            ⅆ                                              α                        1                                                                                    ⅆ                                              α                        2                                                                              )                                            -                              ln                ⁡                                  (                                                            f                      ⁡                                              (                                                  α                          1                                                )                                                                                    f                      ⁡                                              (                                                  α                          2                                                )                                                                              )                                                      )                    ⁢                                                    T                1                            ⁢                              T                2                                                                    T                2                            -                              T                1                                                                        (        5        )            
The so-called DTG signal, calculated as the first derivative from the thermogravimetric signal (weight change signal), has oscillations for modulated measurements. Then it is possible to draw two additional curves: A top curve DTGtop through the peak points of modulated signal DTG, and a bottom curve DTGbottom through the valleys having correspondent values of reaction rates dα2/dt and dα1/dt. If A is the temperature amplitude and T0 is the underlying linear temperature, then T1=T0+A, T2=T0−A, α1=α2, f(α1)=f(α2), and the final expression for activation energy is:
                    Ea        =                  R          *                      ln            ⁡                          (                                                ⅆ                                      α                    2                                                                    ⅆ                                      α                    2                                                              )                                ⁢                                                    T                2                            -              A                                      2              ⁢              A                                                          (        6        )            where the term ln(dα1/dα2) refers to the logarithm of the ratio of minimum and maximum reaction rates. More specifically, it is the amplitude of the logarithm of the first derivative of the thermogravimetric signal indicative of the weight of the sample under investigation at the particular temperature, T. Usually the published curves of activation energy, calculated by this standard method, have minimum point at the highest reaction rate and very high values of activation energy at the beginning and at the end of the reaction (cf. R. L. Blaine and B. K. Hahn, Obtaining Kinetic Parameters by modulated Thermogravimetry, Journal of Thermal Analysis, Vol. 54 (1998) 695-704; and Kun Cheng, William T. Winter, Arthur J. Stipanovic, A modulated-TGA approach to the kinetics of lignocellulosic biomass pyrolysis/combustion, Polymer Degradation and Stability Volume 97, Issue 9, September 2012, Pages 1606-1615). Sometimes these values exceed 1000 kJ/mol for polymers (cf. R. L. Blaine and B. K. Hahn, Obtaining Kinetic Parameters by modulated Thermogravimetry, Journal of Thermal Analysis, Vol. 54 (1998) 695-704), and therefore can not be considered as the activation energy in chemical sense, but rather as the numerical problems at the slow reaction rate. We can see from the above formula (6), that for very slow reaction rate where dα1 is about zero, the calculation of logarithm produces very high uncertainty because the logarithm function changes very fast near zero. Therefore any measurement noise for the almost-zero-signal produces very high error in the activation energy.
Problems in prior art algorithm:                1. The method needs reaction rates a, which are not too close to zero, otherwise the logarithm of ratio can not be found, because at zero the logarithm function goes to negative infinity.        2. The method needs the top curve DTGtop and bottom curve DTGbottom having the same sign to have positive ratio for logarithm calculation. In reality it is not always the case.        3. The positions of the top curve and bottom curve depend on the noise, and therefore noise has big influence on the results.        
If the measured data contain noise, then it is very hard to define correctly the position of the top curve DTGtop and bottom curve DTGbottom. For the unsmoothed data both positions depend on the noise amplitude. But the smoothing of the noise for the signal of unknown shape may distort the shape and reduce the oscillation amplitude of the measured DTG signal. Therefore the smoothing may lead to the evaluation error. Additionally the positions of the top curve and bottom curve for this method can not be determined from the Fourier analysis of only main frequency, because the shape of reaction rate signal can be far from sinus.